ICCG Method Research Articles

Evolving reordering algorithms using an ant colony hyperheuristic approach for accelerating the convergence of the ICCG method

This paper proposes a novel ant colony hyperheuristic approach for reordering the rows and columns of symmetric positive definite matrices. This ant colony hyperheuristic approach evolves heuristics for bandwidth reduction applied to instances arising from specific application areas with the objective of generating low-cost reordering algorithms. This paper evaluates the resulting reordering algorithm in each application area against state-of-the-art reordering algorithms with the purpose of reducing the running times of the zero-fill incomplete Cholesky-preconditioned conjugate gradient method. The results obtained on a wide-ranging set of standard benchmark matrices show that the proposed approach compares favorably with state-of-the-art reordering algorithms when applied to instances arising from computational fluid dynamics, structural, and thermal problems.

ベクトル計算機におけるFill-in付き(M)ICCG法の性能評価

ベクトル計算機におけるFill-in付き(M)ICCG法の性能評価

A novel CN–ICCG–FDTD algorithm research of plasma reflection and transmission characteristics of electromagnetic wave

The advantage of the ICCG method for solving large sparse matrix is taken in the CN–FDTD equation solving. The CN–ICCG–FDTD can accelerate iteration for numerical calculation, and reduce memory overhead. Maxwell's equation of the electromagnetic wave in dispersive medium plasma is deduced from the CN format after being differentiated in the time domain, the FDTD method become unconditionally stable. In this process, a large sparse matrix is generated. As for solving such matrix, ICCG method has sufficient advantages. In ICCG method, convergence is so fast and stable that it is quite easy for computer programming. In addition, data required by ICCG method are memorized in a computer in the format of the one-dimensional compression, which helps the computer memory save a large amount of capacity, especially when the issues are rather complex to cope with, this algorithm will display such strength. With these advantages, ICCG method is able to calculate the reflection coefficient and the transmission coefficient of electromagnetic waves in plasma as well as the phase angle of them. The calculation indicates that the time step required by CN–ICCG–FDTD method has eliminated the constraints of the CFL, thus shortening the required time, making the calculated result stable and accurate, and improving the efficiency of programming.

A research on the CN-ICCG–FDTD algorithm of plasma photonic crystals and the transmission coefficient of electromagnetic wave

The general FDTD method cannot calculate a complex object because of limit of the CFL conditions. CN-FDTD is an improvement on the FDTD method, and it can get rid of the CFL conditions and become wholly unconditional stable form. The advantage of the ICCG method for solving large sparse matrix will be taken in the CN-FDTD equation solving. ICCG method can accelerate iteration for numerical calculation, reduce memory overhead and be easy to be programmed at the same time. CN-ICCG–FDTD method researches the object model of plasma photonic crystals. Plasma photo crystals have unique properties. They can show mainly characteristics of plasma, or have the main properties of photonic crystals. Combination of plasma and photonic crystals makes it possible to adjust the plasma parameters to control the properties of photonic crystals. In this paper, it calculates characteristic parameters of plasma photonic crystals under different conditions, such as the reflection electric field, the transmission electric field, the transmission coefficient, to verify accuracy of CN-ICCG–FDTD method. The results of this paper prove the method is accurate, performs stably, and has certain advantages.

On the Practical Implementation of ICCG Method Solving Real Symmetric Linear Equation Arising in Edge-based Finite Element Analysis

On the Practical Implementation of ICCG Method Solving Real Symmetric Linear Equation Arising in Edge-based Finite Element Analysis

A New Acceleration Factor Decision Method for ICCG Method Based on Condition Number

The ICCG method is widely used to solve a sparse symmetric linear system which results from the finite element method. In order to improve the convergence property of the ICCG method, the introduction of the acceleration factor was proposed. The automatic acceleration factor decision method, using the incomplete Cholesky decomposition of a coefficient matrix, has been previously proposed. However, when employing the previously proposed automatic decision method, much more iterations of ICCG method are sometimes necessary, compared with using the optimum acceleration factor, which minimizes the number of ICCG iterations. In this paper, we propose a new acceleration factor decision method. The proposed method takes into account the condition number of the coefficient matrix. It is well known that the condition number represents the quality of the matrix, so the optimum acceleration factor should be decided to minimize the condition number of the coefficient matrix. However, the condition number of the coefficient matrix is not almost available due to requiring a large memory in computing. Therefore, we develop a new method using submatrix, that does not need a large memory. The procedure and demonstration of the proposed method are described in this paper.

Orthogonalized Infinite Edge Element Method—Convergence Improvement by Orthogonalization of Hilbert Matrix in Infinite Edge Element Method

This paper proposes an orthogonalization of the Hilbert matrix in element matrices of the infinite edge elements. The validity of the infinite edge element is demonstrated in previous researches, but the Hilbert matrix results in extremely slow convergence in the ICCG method, especially when using higher order expansions. The proposed orthogonalization technique improves the convergence drastically and it makes the infinite elements practical in the electromagnetic FEM analysis of the open boundary problems in quasi-static magnetic fields.

Algebraic multigrid method for 3D DC resistivity modelling

Complete text of publication follows. Despite the seemingly ever growing power of computers, inversions of full 3D DC resistivity data are still challenging and time-consuming. Since the inversion processes are based on forward modeling, the speed up of the 3D DC resistivity modeling processes is always an interesting topic. As an iterative solver for discretized large linear equation systems, Multigrid (MG) methods are known for their high convergence rate which is independent to the number of grid nodes. However, when applied to 3D DC resistivity modeling, this attractive property may be affected by high conductivity contrasts. Algebraic Multigrid (AMG) method, based on matrix-dependent operators, can be used to retrieve this problem. The equation of continuity with mixed boundary conditions for 3D DC resistivity modelling is discretized using a standard 7-point finite difference scheme on the finest level with non-equidistant grids. Secondary potential approach is used to remove the singularities on the right-hand-side. Some forward modelling calculations are carried out with an AMG code named AMG1R5, which was made publically available in the mid-1980s. Three underground models, a three-layer model, a vertical contact model and a cubic model, are investigated. The affections of grid sizes and conductivity contrasts to the convergence rate and computational costs are under investigation. Comparing with the ICCG method, a widely used iterative method in 3D DC resistivity modeling, AMG method shows its superior performance just as expected.

Molecular structural mechanics approach to carbon nanotubes on graphics processing units

A molecular structural mechanics approach to carbon nanotubes on graphics processing units (GPUs) is reported. As a powerful parallel and relatively low cost processor, the GPU is used to accelerate the computations of the molecular structural mechanics approach. The data structures, matrix-vector multiplication algorithm, texture reduction algorithm, and ICCG method on the GPU are presented. The computations for Young's moduli of carbon nanotubes by the molecular structural mechanics approach on the GPU show its accuracy. The running times of large degree of freedom (DOF) carbon nanotubes, whose DOF is larger than 100,000, on the GPU are compared against those on the CPU , proving the GPU can accelerate the computations of the molecular structural mechanics approach to carbon nanotubes.

Investigation of Algebraic Multigrid Method for Magnetic Field Analysis of an Electric Machine With a Laminated Core

In this paper, we describe the applicability of algebraic multigrid method (AMG method) as a fast solution for magnetic field analysis of an electric machine with a laminated core. In an experimental analysis, we have found that there are some cases where the calculation speed can be improved by using the AMG method for a fine mesh that models a laminated iron core exactly. Therefore we investigated the convergence characteristic of the standard incomplete Cholesky conjugate gradient method (standard ICCG method) and the conjugate gradient method preconditioned with AMG method (AMGCG method) for the simple and actual analysis models. The numerical results show that the AMGCG method provides fast solutions for laminated mesh models for which the standard ICCG method does not work well in many analysis cases.

New multigrid method including elimination algorithm based on high‐order vector finite elements in three‐dimensional magnetostatic field analysis

AbstractA new multigrid method based on high‐order vector finite elements is proposed in this paper. Low‐level discretizations in this method are obtained by using low‐order vector finite elements for the same mesh. The Gauss–Seidel method is used as a smoother, and a linear equation of lowest level is solved by the ICCG method. But it is often found that multigrid solutions do not converge into ICCG solutions. An elimination algorithm of constant term using a null space of the coefficient matrix is also described. In three‐dimensional magnetostatic field analysis, convergence time and number of iteration of this multigrid method are discussed with the conventional ICCG method. © 2009 Wiley Periodicals, Inc. Electron Comm Jpn, 92(1): 39–45, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecj.10228

POWER GRIDS ANALYSIS IN COMPRESSED KRYLOV-SUBSPACE METHODS

An improved method is proposed based on compressed and Krylov-subspace iterative approaches to perform efficient static and transient simulations for large-scale power grid circuits. It is implemented with CG and BiCGStab algorithms and an excellent result has been obtained. Extensive experimental results on large-scale power grid circuits show that the present method is over 200 times faster than SPICE3 and around 10–20 times faster than ICCG method in transient simulations. Furthermore, the presented algorithm saves the memory usage over 95% of SPICE3 and 75% of ICCG method, respectively while the accuracy is not compromised.

Evaluation Index of Acceleration Factor and Ordering in Shifted ICCG Method for Electromagnetic Field Analyses

This paper introduces a simple way to evaluate the preconditioning effect in the shifted ICCG method. The proposed evaluation index is easily calculated with little additional memory in an incomplete factorization process; thus, it can be used for setting appropriate parameters for the solver, such as an acceleration factor and the ordering of unknowns, prior to the time-consuming iteration process. Numerical results show that the evaluation index gives the good estimate of the number of iterations necessary for convergence

New Multigrid Method Including Elimination Algolithm Based on High-Order Vector Finite Elements in Three Dimensional Magnetostatic Field Analysis

A new multigrid method based on high-order vector finite elements is proposed in this paper. Low level discretizations in this method are obtained by using low-order vector finite elements for the same mesh. Gauss-Seidel method is used as a smoother, and a linear equation of lowest level is solved by ICCG method. But it is often found that multigrid solutions do not converge into ICCG solutions. An elimination algolithm of constant term using a null space of the coefficient matrix is also described. In three dimensional magnetostatic field analysis, convergence time and number of iteration of this multigrid method are discussed with the convectional ICCG method.

FEM Computation of Magnetic Fields in Anisotropic Magnetic Materials

The magnetic fields in nonlinear anisotropic magnetic materials were analyzed by using the Finite Element Method (FEM). The measured data was directly used in the computation without a complicateded smoothing. The resultant asymmetric linear equations were solved by using the ILUBiCGStab method without symmetrization or the ICCG method with symmetrization. The magnetic flux distributions in a ring core model showed the characteristic patterns according to the non-oriented, grain-oriented and doubly-oriented magnetic properties. The good convergence of the Newton-Raphson nonlinear iteration was attained by the iterative solvers without special techniques for the smoothing.

322 重合メッシュ問題に対する新しい並列化技法AMB法によるICCG法について(3.X-FEM,S-FEM,計算破壊力学の最新展開(計算力学手法とその基礎理論と応用,実験/計算ハイブリッドテクニック,き裂先端変形場),OS21 計算破壊力学の最新展開(計算力学手法とその基礎理論と応用,実験/計算ハイブリッドテクニック,き裂先端変形場))

重合メッシュ問題から生じた行列に対して,従来の並列化技法であるABRB法や高速版のPBFRIC分解では十分な並列度数の確保ができないことがあった.そこで,新しい代数マルチブロック(Algebraic Multi Block:以下,AMBと略す)法によるICCG法の並列化を行った.

Parallel iterative solvers for finite-element methods using an OpenMP/MPI hybrid programming model on the Earth Simulator

An efficient parallel iterative method for finite-element method has been developed for symmetric multiprocessor (SMP) cluster architectures with vector processors such as the Earth Simulator. The method is based on a three-level hybrid parallel programming model, including message passing for inter-SMP node communication, loop directives by OpenMP for intra-SMP node parallelization and vectorization for each processing element (PE). Simple 3D linear elastic problems with more than 2.2 × 10 9 DOF have been solved using 3 × 3 block ICCG(0) method with additive Schwarz domain decomposition and PDJDS/CM-RCM reordering on 176 nodes of the Earth Simulator, achieving performance of 3.80 TFLOPS. Furthermore, effect of color number in reordering has been evaluated on various types of computers.

Three-level hybrid vs. flat MPI on the Earth Simulator: Parallel iterative solvers for finite-element method

An efficient parallel iterative method for finite element method has been developed for symmetric multiprocessor (SMP) cluster architectures with vector processors such as the Earth Simulator. The method is based on a three-level hybrid parallel programming model, including message passing for inter-SMP node communication, loop directives by OpenMP for intra-SMP node parallelization and vectorization for each processing element (PE). Simple 3D linear elastic problems with more than 2.2 × 10 9 DOF have been solved using 3 × 3 block ICCG(0) method with additive Schwarz domain decomposition and PDJDS/CM-RCM reordering on 176 nodes of the Earth Simulator, achieving performance of 3.80 TFLOPS. The PDJDS/CM-RCM reordering method provides excellent vector and parallel performance in SMP nodes. A three-level hybrid parallel programming model outperforms flat MPI in the problems involving large numbers of SMP nodes.

重合メッシュ法で現れる線形方程式に対する対角緩和つき準ロバストICCG法の収束性(OS8c メッシュフリー/粒子法)

重合メッシュ法で現れる線形方程式に対する対角緩和つき準ロバストICCG法の収束性(OS8c メッシュフリー/粒子法)

Parallelization of ICCG Method Using Three Dimensional Block Red-Black Ordering Method

The present paper deals with parallelization of ICCG method. A reordering (parallel ordering) technique is one of typical parallel processing techniques of ICCG method. While several parallel orderings have been proposed, the block red-black ordering that is proposed by the authors is relatively new ordering. This ordering method has advantages in convergence rate and in synchronization costs, and its effectiveness was confirmed in a preliminary numerical test for 2-d finite difference analyses. In the present paper, this method is enhanced for 3-d finite difference analyses, which require more computational efforts than 2-d analyses. We here describe the details of the implementation procedure of the block red-black ordering in 3-d analyses. Numerical tests for a 3-d partial differential equation problem are performed. It is shown that 3-d block red-black ordering method attains high parallel efficiency due to fast convergence and effective use of data cache.